As for the terminology used in this paper we follow [1]} unless stated explicitly otherwise.
In particular, the subset \(E(v)\subseteq E(G)\)  denotes the set of edges incident to \(v\in V(G)\) .
For definitions we refer to [2]} but for completeness' sake we repeat some of them. Moreover, we present several known results (Theorems A-E) which can be viewed as the frame in which the results of this paper take place.
